Hans Duistermaat’s contributions to Poisson geometry

Sjamaar, Reyer

doi:10.1007/s00574-011-0035-2

Cited by 4 publications

(5 citation statements)

References 43 publications

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“…Remarkably, the lower version of the theory, the symplectic story, i.e. symplectic T -torsors, as well as its relevance to Lagrangian fibrations, has already appeared in [60]; v3: Gerbes over orbifolds: although a large part of our discussion will be carried out in the case where B is a smooth manifold, in general our leaf spaces B are orbifolds. The passage from manifolds to orbifolds will be based again on Haefliger's philosophy (Remark 2.3.3).…”

Section: Symplectic Gerbes Over Manifoldsmentioning

confidence: 99%

“…The construction of the Chern class class has a natural symplectic version, that dates back to Duistermaat's work on global action-angle coordinates [28]. This was further clarified and generalized by Delzant and Dazord [26] and Zung [67], and rephrased in the language of symplectic torsors by Sjamaar [60]. The relevant sheaf is no longer T , but rather the subsheaf T Lagr of local Lagrangian sections:…”

Section: A Symplectic Version Of the Theorymentioning

confidence: 99%

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Poisson manifolds of compact types (PMCT 1)

Crainic

Fernandes

Torres

2017

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Abstract. This is the first in a series of papers dedicated to the study of Poisson manifolds of compact types (PMCTs). This notion encompasses several classes of Poisson manifolds defined via properties of their symplectic integrations. In this first paper we establish some fundamental properties and constructions of PMCTs. For instance, we show that their Poisson cohomology behaves very much like the de Rham cohomology of a compact manifold (Hodge decomposition, non-degenerate Poincaré duality pairing, etc.) and the Moser trick can be adapted to PMCTs. More important, we find unexpected connections between PMCTs and Symplectic Topology: PMCTs are related with the theory of Lagrangian fibrations and we exhibit a construction of a nontrivial PMCT related to a classical question on the topology of the orbits of a free symplectic circle action. In subsequent papers, we will establish deep connections between PMCTs and Integral Affine Geometry, Hamiltonian G-spaces, Foliation Theory, orbifolds, Lie Theory and symplectic gerbes.

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Section: Symplectic Gerbes Over Manifoldsmentioning

confidence: 99%

Section: A Symplectic Version Of the Theorymentioning

confidence: 99%

Poisson manifolds of compact types (PMCT 1)

Crainic

Fernandes

Torres

2017

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…From this we see that the canonical Hamiltonian stratification of the orbit space of the Hamiltonian G-space T * (G/H) has six strata, three of which correspond to the semialgebraic submanifolds of (71) given by the respective intersections of ( 71) with {x 1 < 0}, {x 1 = 0} and {x 1 > 0}, and the other three of which correspond to the semi-algebraic submanifolds of (72) given by:…”

Section: 23mentioning

confidence: 99%

“…As for toric manifolds, the momentum map J of a toric (T , Ω)-space is an integrable system in local coordinates for M . that associates to [J : (S, ω) → M ] its so-called Lagrangian Chern class (called Lagrangian class in [72,86]). The key insight leading to (87) is that fibrations as in (87) can be viewed as certain principal T Λ -bundles, called symplectic (T Λ , Ω Λ )-torsors in [15,72].…”

Section: 33mentioning

confidence: 99%

“…that associates to [J : (S, ω) → M ] its so-called Lagrangian Chern class (called Lagrangian class in [72,86]). The key insight leading to (87) is that fibrations as in (87) can be viewed as certain principal T Λ -bundles, called symplectic (T Λ , Ω Λ )-torsors in [15,72]. Indeed, any such fibration is a Poisson map into (M, 0) that admits a (necessarily unique) Hamiltonian (T Λ , Ω Λ )-action along it, which turns it into a symplectic (T Λ , Ω Λ )-torsor (cf.…”

Section: 33mentioning

confidence: 99%

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Hamiltonian actions of compact type from a Poisson point of view

Mol¹

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Contents Acknowledgements A brief introduction Part 1. Stratification of the transverse momentum map Introduction 1. The normal form theorem 1.1. Background on Hamiltonian groupoid actions 1.2. The local invariants 1.3. The local model 1.4. The proof 1.5. The transverse part of the local model 2. The canonical Hamiltonian stratification 2.1. Background on Whitney stratifications of reduced differentiable spaces 2.2. The stratifications associated to Hamiltonian actions 2.3. The regular parts of the stratifications 2.4. The Poisson structure on the orbit space 2.5. Symplectic integration of the canonical Hamiltonian strata Part 2. Toric actions of regular and proper symplectic groupoids Introduction 3. The momentum image and the ext-invariant 3.1. Toric representations of infinitesimally abelian compact Lie groups 3.2. Delzant subspaces of integral affine orbifolds 3.3. The ext-invariant and the ext-sheaf 3.4. A normal form on invariant neighbourhoods 3.5. The remaining proofs 4. The structure theorems and the splitting theorem 4.1. Constructing a natural toric (T , Ω)-space out of a Delzant subspace 4.2. The sheaf of automorphisms and the sheaf of invariant Lagrangian sections 4.3. Proof of the structure theorems 4.4. Proof of the splitting theorem A. Poisson geometric characterization of toric actions B. Background on manifolds with corners C. A vanishing result for the second structure group Index Bibliography Curriculum Vitae Samenvatting (ook voor niet-wiskundigen)Using the proposition below, the short exact sequence (19) translates into the short exact sequence (17), whereas (18) translates into (16). In particular, this proves part b. □ Proposition 1.12. Let (G, Ω) ⇒ M be a symplectic groupoid and suppose that we are given a left Hamiltonian (G, Ω)-action along J : (S, ω) → M . Further, let p ∈ S.a) The symplectic orthogonal (13) of the tangent space T p O to the orbit O through p coincides with Ker(dJ p ). b) The isotropy Lie algebra g p , viewed as subset of T *x M via (117), is the annihilator of Im(dJ p ) in T x M , where x = J(p). This is readily derived from the momentum map condition (6).1.2.3. The symplectic normal representation. Notice that the symplectic form ω on S descends to a linear symplectic form ω p on the symplectic normal space (12). Proposition 1.13. (SN p , ω p ) is a symplectic G p -representation.Proof. We ought to show that ω p is G p -invariant. Note that, for any v ∈ Ker(dJ p ) and g ∈ G p :g. So, using Proposition 1.12a we find that for all v, w ∈ T p O ω and g ∈ G p :where in the last step we applied (5). □ Definition 1.14. Given a Hamiltonian action as above, we call (20) (SN p , ω p ) ∈ SympRep(G p ) its symplectic normal representation at p.

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